1. Solutions and Thermobarometry
Lecture 12

2. Plagioclase Solution
Unlike alkali feldspar, Na-Ca feldspar (plagioclase) forms a complete solid (and liquid) solution.
Let’s construct the melting phase diagram from thermodynamics.
For simplicity, we assume both liquid and solid solutions are ideal.

3. Plagioclase Solution Condition for equilibrium: Chemical potential is
e.g.:
Chemical potential is
Combining these:
standard states are the pure end member solids and liquids.

4. error in book: Ab on lhs should be An
Plagioclase Solution
The l.h.s. is simply ∆Gm for the pure component:
rearranging
Since XAn = 1 - XAb
error in book: Ab on lhs should be An

5. Plagioclase Solution From:
We can solve for mole fraction of Ab in the liquid:
The mole fraction of any component of any phase in this system can be predicted from the thermodynamic properties of the end-members.
In the ideal case, as here, it simply depends on ∆Gm and T.
In a non-ideal case, it would depend on Gexcess as well.
Computing the equation above (and a similar one for the solid), we can compute the phase diagram.

6. We now have enough thermodynamics to put it to some real use: calculating the temperatures and pressures at which mineral assemblages (i.e., rocks) equilibrated within the Earth.

7. Some theoretical considerations
We have seen that which phase assemblage is stable and the composition of those phases depends on ∆Gr, which we use to calculate K
We also know ∆Gr depends on T and P.
Reactions that make good geothermometers are those that depend strongly on T.
What would characterize a good geothermometer?
A good geobarometer would be one where K depends strongly depending on P
A good geothermometer will have large ∆H; a good geobarometer will have large ∆V.

8. Univariant Reactions
Univariant (or invariant) reactions provide possible thermobarometers.
There are 3 phases in the Al2Si2O5 system.
When two coexist, we need only specify either T or P, the other is then fixed.
All three can coexist at just one T and P.
First is rare, second is rarer.

9. Garnet Peridotite Geobarometry
Garnet becomes the high pressure aluminous phase in the mantle, replacing spinel.
Aluminum also dissolves in the orthopyroxene (also clinopyroxene)
We can write the reaction as:
Mg2Si2O6+MgAl2SiO6 = Mg3Al2Si3O12
l.h.s. is the opx solid solution - Al end member does not exist as pure phase.
Significant volume change associated with this reaction (but also depends on T).
Other complexities arise from Ca, Fe, and Cr in phases.
Original approach of Wood and Banno generally assumed ideal solution

10. Garnet Peridotite Geobarometry
Subsequent refinements used asymmetric solution model to match experimental data.
Recognize two distinct sites in opx crystal:
Smaller M1: Al substitutes here
Larger M2: Ca substitutes here
P given by
where C3 is constant and other parameters depend on K, T, and composition.

11. Solvus Equilibria
Another kind of thermobarometer is based on exsolution of two phases from a homogenous single phase solution.
This occurs when the excess free energy exceeds the ideal solution term and inflections develop, as in the alkali feldspar system.
Because it is strongly temperature dependent and not particularly pressure dependent, this makes a good geothermometer.

12. Temperature in Peridotites
Ca2+ 
Temperatures calculated from compositions of co-existing orthopyroxene (enstatite) and clinopyroxene (diopside) solid solutions, which depend on T.

13. Exchange Reactions
There are a number of common minerals where one or more ions substitutes for others in a solid solution.
The Fe2+–Mg2+ substitution is common in ferromagnesian minerals.
Let’s consider the exchange of Mg and Fe between olivine and a melt containing Mg and Fe.
This partitioning of these two ions between melt and olivine depends on temperature.
We can use a electron microprobe to measure the composition of olivine and co-existing melt (preserved as glass).

14. Olvine-Melt Geothermometer
Reaction of interest can be written as:
MgOol + FeOl = MgOl + FeOol
(note, this does not involve redox, so we write it in terms of oxides since these are conventionally reported in analyses. We could write it in terms of ions, however.)
Assuming both solid and liquid solutions are ideal, the equilibrium constant for this reaction is:
Unfortunately ∆H for the reaction above is small, so it has weak temperature dependence.

15. Roeder & Emslie Geothermometer
Roeder & Emslie (1970) decided to consider two separate reactions:
MgOliq –> MgOOl and FeOliq –> FeOOl
Based on empirical data, they deduced the temperature dependence as:
and
See Example 4.3 for how to do the calculation - biggest effort is simply converting wt. percent to mole fraction.

16. Buddington and Lindsley Oxide Geothermometer
Things get interesting in real systems containing Ti, because both magnetite and hematite are solid solutions.
Partition of Fe and Ti between the two depends on T and ƒO2.
Recall this diagram from Chapter 3
rutile

17. Magnetite & Ilmenite Magnetite & Ilmenite at high T in a gabbro
Ilmenite exsolving from magnetite at low T

18. Buddington and Lindsley Oxide Geothermometer
The reaction of interest is:
yFe2TiO4 + (1-y)Fe3O4 + ¼O2 = yFeTiO3 + (3/2 -y)Fe2O3
magnetite s.s. hematite s.s.
The equilibrium constant for this reaction is
The reaction can be thought of as a combination of an exchange reaction:
Fe3O4 + FeTiO3 = Fe3TiO4 + Fe2O3
magnetite + illmenite = ulvospinel + hematite
plus the oxidation of magnetite to hematite:
4Fe3O4 + O2 = 6Fe2O3

19. Computing Temperature and Oxygen Fugacity
The calculation is complex because the system cannot be treated as ideal (except titanomagnetite above 800˚C). Equilibrium constant is:
and
Must calculate λ’s using asymmetric solution model (using interaction parameters), then solve for T and ƒO2. Example 4.4 shows how.

20. Update
There have been a number of revisions to the Fe-Ti oxide geothermobarometer since the work of Buddington and Lindsley.
One of the most recent is by Ghiorso and Evans (2008). This is far more sophisticated and takes account of crystal structure and the specific sites in the crystal lattices where substitution occurs.
This allows more accurate estimate of T and fO2, but is computationally far more complex.
They have an online calculator at:
From Ghiorso and Evans (2008).

22. Next up: MELTS modelling
Visit the MELTS web page to find out more about this software. Several possible downloads from here, including Rhyolite-Melts (for Mac, but you need an X-Windows system) and Rhyolite_MELTS for Excel (Windows only).
We will run some examples on this site.
Visit the CalTech magmasource web site:
Read about installing alphaMELTS.
Download and install the latest version (1.5) and any necessary virtual machine program (for Windows).